I came across an exercise that says: Find $\displaystyle \int S(x) dx$, where $S(x)$ is $\displaystyle \int_0^x \sin(\pi t^2/2)dt$
I understand what I should get $\displaystyle\int\left( \int_0^x \sin(\pi t^2/2)dt\right)dx$
But I don't know if the integral of fresne is known (I think not because there are numerical approximations)
Can someone guide me? Maybe my interpretation of the problem is wrong, or how can I attack it?
As stated in the answer by @GEdgar, you should integrate by parts. To make the computation easier, evaluate $\int S(x)\,dx$ directly instead of writing out the definite integral. You can use the fundamental theorem of calculus to evaluate $S'(x)$ provided that you choose $u=S(x)$.
Let $u=S(x)$ and $dv=dx$. Then $du = S'(x)\,dx$ and $v=x$ so integrating by parts gives
$$\int S(x)\,dx = xS(x)-\int x \sin\left(\frac{\pi x^2}{2}\right)\,dx.\tag{1}$$
To evaluate the integral on the right, substitute $u = (\pi x^2)/{2} \implies du= \pi xdx$ to find
$$\int x\sin\left(\frac{\pi x^2}{2}\right)\,dx=\frac{1}{\pi}\int\sin(u)\,du=-\frac{\cos(u)}{\pi}+C=-\frac 1\pi\cos\left(\frac{\pi x^2}{2}\right)+C.\tag{2}$$
Insert $(2)$ into the integral on the RHS of $(1)$ to conclude.